Concatenating multiple columns into one while copying values of other columns

Using melt with dropna

yourdf=df.melt(['Date','DV']).dropna()
yourdf
          Date   DV variable            value
0   22/02/2019  200      FA1           Lazard
1    2/02/2019   50      FA1         Deutsche
2   22/02/2019  120      FA1  ChinaSecurities
4    2/02/2019   50      FA2            Ondra
5   22/02/2019  120      FA2           Ballas
8   22/02/2019  120      FA3            Daiwa
11  22/02/2019  120      FA4    MorganStanley

Using stack:

df = (df.set_index(['Date','DV']).stack()
        .reset_index(level=[0,1], name='FA')
        .reset_index(drop=True))

print(df)
         Date   DV                FA
0  22/02/2019  200            Lazard
1   2/02/2019   50          Deutsche
2   2/02/2019   50             Ondra
3  22/02/2019  120  China Securities
4  22/02/2019  120            Ballas
5  22/02/2019  120             Daiwa
6  22/02/2019  120    Morgan Stanley