I think I would do this with two groupbys.
First to calculate the “weighted average”:
In [11]: g = df.groupby('Date')
In [12]: df.value / g.value.transform("sum") * df.wt
Out[12]:
0 0.125000
1 0.250000
2 0.416667
3 0.277778
4 0.444444
dtype: float64
If you set this as a column, you can groupby over it:
In [13]: df['wa'] = df.value / g.value.transform("sum") * df.wt
Now the sum of this column is the desired:
In [14]: g.wa.sum()
Out[14]:
Date
01/01/2012 0.791667
01/02/2012 0.722222
Name: wa, dtype: float64
or potentially:
In [15]: g.wa.transform("sum")
Out[15]:
0 0.791667
1 0.791667
2 0.791667
3 0.722222
4 0.722222
Name: wa, dtype: float64
