You can try to cat the files into one using
cdo cat V?.nc combined.nc
and then make a new variable using the expression command
intercept=2.3123 # or whatever the value is
cdo expr,'y=1 / (1 + exp(-(${intercept} + V1 * 1.87 + V2 * 1.53 - 4.18*V3 + V4 * 3.58 + V5 * 9.13)))' combined.nc newvar.nc
I'm assuming the variable in V1.nc is called "V1" etc, if not substitute the name accordingly.
EDIT: According to the comment each file uses the same variable name. So either the variable should be renamed in each file using NCO prior to the above, or one could combine the original files using separate cdo functions as follows:
cdo -pow,-1 -addc,1 -exp -mulc,-1 -addc,$intercept -add -mulc,1.87 V1.nc -add -mulc,1.53 V2.nc -add -mulc,-4.18 V4.nc -add -mulc,3.58 V4.nc -mulc,9.13 V5.nc mynewvariable.nc
I've used piping to get it on one line and avoid intermediate files, check my equation carefully but I hope it doesn't contain mistakes. With piping it helps to read from right to left.
The variable in the file mynewvariable.nc will also be called Band1; you might want to change the metadata appropriately using nco, always a good habit to have even if the files are only for your own use...